## Abstract

A laser phase and frequency noise measurement method by an unbalanced Michelson interferometer composed of a 3 × 3 optical fiber coupler is proposed. The relations and differences of the power spectral density (PSD) of differential phase and frequency fluctuation, PSD of instantaneous phase and frequency fluctuation, phase noise and linewidth are derived strictly and discussed carefully. The method obtains the noise features of a narrow linewidth laser conveniently without any specific assumptions or noise models. The technique is also used to characterize the noise features of a narrow linewidth external-cavity semiconductor laser, which confirms the correction and robustness of the method.

© 2015 Optical Society of America

## 1. Introduction

Single frequency narrow-linewidth lasers are fundamental to a vast array of applications in fields including metrology, coherent optical communications, high resolution sensing, and LIDAR [1,2]. In these applications, the phase and frequency noise is one of the key factors to affect the system performance. The characterization and measurement of the phase and frequency noise are very important for the applications, and thus have been one of the most attractive subjects of researches in laser and photonics field. The phase and frequency noise of such lasers can be conveniently described either in terms of linewidth or in terms of the power spectral density (PSD) of their phase or frequency noise. The linewidth gives a basic and concise parameter for characterizing laser coherence, but lacks of detailed information on frequency noise and its Fourier frequency spectrum, which is needed for understanding the noise origins and improving laser performances. Therefore the measurement of frequency noise PSD is a focus of attention in the field, especially for lasers of very high coherence, whose linewidth is not easy to be measured.

To measure the phase and frequency noise, many methods have been proposed, such as beat note method [3], recirculating delayed self-heterodyne (DSH) method [4], DSH technique based on Mach-Zehnder interferometer with 2 × 2 coupler [5,6] or Michelson interferometer (MI) with 2 × 2 coupler [7]. These methods can obtain good measurement results but need some strict conditions. The beat note method needs a high coherent source as a reference. The recirculating DSH method needs very long fiber delay lines. The DSH interferometers with 2 × 2 coupler need to control of quadrature point by some active feedback methods and accurate calibration.

To overcome these difficulties, we introduce a robust technique that can demodulate directly the laser differential phase accumulated in a delay time, and then derive strict mathematical relations between the laser differential phase and the laser phase noise or frequency noise that can describe the complete information on laser phase and frequency noise. Because 3 × 3 optical fiber coupler acts as a 120-degree optical hybrid, it can demodulate the differential phase of the input light and has been used for DxPSK signal demodulation [8], optical sensors [9], optical field reconstruction and dynamical spectrum measurement [10]. In this letter, an unbalanced Michelson interferometer composed of a 3 × 3 optical fiber coupler and two Faraday rotator mirrors is utilized to demodulate the differential phase of a laser. The structure has the advantages of polarization insensitive and adjust-free. Especially, it doesn’t need any active controlling operation that is used in the DSH methods with 2 × 2 coupler. Furthermore, based on the differential phase and strict physical and mathematical derivation, the PSD of the differential phase fluctuation and frequency fluctuation, the PSD of the instantaneous phase fluctuation and frequency fluctuation, laser phase noise and linewidth are completely calculated and discussed.

## 2. Experimental setup

The experimental setup is shown in Fig. 1. It consists of a commercially available 3 × 3 optical fiber coupler (OC), a circulator (C), two Faraday rotator mirrors (FRMs), three photodetectors (PDs), a data acquisition board (DAC) or a digital oscilloscope, and a computer. An unbalanced Michelson interferometer is composed of the 3 × 3 coupler and the FRMs. The FRM will remove the polarization fading of interferometer, caused by external disturbance on its two beam fibers. The laser under test (LUT) injects the left port 1 of the 3 × 3 optical fiber coupler through a circulator and then splits into 3 parts by the coupler. Two of them interfere mutually in the coupler after reflected by Faraday mirrors and with different delay times, and the third part of them is made reflection-free. Then the interference fringes are obtained from the left port 1, 2, 3 of the coupler, and read by a DAC or a digital oscilloscope.

On the other hand, the length or index of the fiber configuring the interferometric arms would change randomly because of temperature fluctuations, vibration and other types of the environmental disturbances, thus it induces low frequency random phase drifts in the interferometric signal. So in our experimental setup, the complete interferometer is housed in an aluminum box enclosed in a polyurethane foam box for thermal and acoustic isolation. Meantime, the two fiber arms of the MI are placed closely in parallel to improve the stability against the perturbation.

Following the derivation in [8], by using of the three readings *I _{n}*(

*t*) (

*n*= 1, 2, 3), the differential phase fluctuation Δ

*φ*(

*t*) of LUT accumulated in delay time

*τ*is expressed as:

*φ*(

*t*) is the instantaneous phase fluctuation of LUT at time

*t*,

*τ*is the delay time difference of two interference arms,

*X′*

_{1}(

*t*) and

*X′*

_{2}(

*t*) are expressed as:

*η*= 2

_{n}*r*

_{n}c_{n}_{1}

*c*

_{n}_{2}cos(

*θ′*

_{n}_{2}

*+ θ*

_{21}-

*θ′*

_{n}_{1}

*-θ*

_{11}),

*ς*= 2

_{n}*r*

_{n}c_{n}_{1}

*c*

_{n}_{2}sin(

*θ′*

_{n}_{2}

*+ θ*

_{21}-

*θ′*

_{n}_{1}

*-θ*

_{11}),

*ξ*=

_{n}*r*[(

_{n}*c*

_{n}_{1})

^{2}+ (

*c*

_{n}_{2})

^{2}],

*c*=

_{nm}*p*

_{mm}b′_{nm}b_{m}_{1},

*n*= 1, 2, 3 is the output port number of MI, and

*m*= 1, 2 is the interference arm port number.

*p*is the transmission loss of the interference arm,

_{mm}*b*and

_{mn}*θ*are the splitting ratio and phase delay from the port

_{mn}*n*to port

*m*of coupler,

*b′*and

_{mn}*θ′*are the splitting ratio and phase delay from the port

_{mn}*m*to port

*n*of coupler,

*r*is the responsivity of PDs. So the parameters

_{n}*η*,

_{n}*ς*,

_{n}*ξ*are constant for the setup once the devices and structure are determined. They can be obtained by a broadband light source without measuring each parameter one by one as said in [8]. In our setup, a swept laser source with linewidth of about 2.5 kHz [11] is used as the broadband light source to show clearly the small free spectral range (FSR) of the MI. The measured interference fringe is shown in the inset figure of Fig. 1. On the other hand, all parameters of the devices are considered in the differential phase fluctuation calculation process, so the possible errors from device defects like imperfect splitting ratio or phase difference are removed and the requirements for the device performance parameters are also relaxed. In our experimental setup, the final setup parameters are

_{n}*τ*= 244 ns (corresponding FSR of the MI is 4.1 MHz),

*η*

_{1}= 0.1198,

*η*

_{2}= −0.0519,

*η*

_{3}= −0.0634,

*ς*

_{1}= 0.0002,

*ς*

_{2}= 0.1366,

*ς*

_{3}= −0.1094,

*ξ*

_{1}= 0.1223,

*ξ*

_{2}= 0.1491,

*ξ*

_{3}= 0.1290.

Then considering the relation between the delay phase *φ* and frequency *ν*, *φ* = 2π*nlν*/c = 2π*τν*, the differential phase variation is from the laser frequency variation in the time interval *τ*, because the delay time difference *τ* of the MI is fixed and the random variation of the fiber is eliminated carefully by some techniques as described above. So from the differential phase fluctuation Δ*φ*(*t*), the laser frequency fluctuation in time interval *τ* defined as differential frequency fluctuation Δ*ν*(*t*) can be expressed as:

Hence, the PSD of differential phase fluctuation Δ*φ*(*t*) and differential frequency fluctuation Δ*ν*(*t*) can be calculated respectively in the computer by PSD estimation method [12], and denoted as *S*_{Δ}* _{φ}*(

*f*) and

*S*

_{Δ}

*(*

_{ν}*f*), respectively, where

*f*is the Fourier frequency. Meantime, from the linearity of the Fourier transform, they have a fixed relation:

So far, the differential phase fluctuation Δ*φ*(*t*) accumulated in the delay time difference *τ* of the MI, corresponding differential frequency fluctuation Δ*ν*(*t*) and their PSD are calculated. But these values are not the instantaneous information of the LUT. Furthermore, considering the relation of differential phase fluctuation Δ*φ*(*t*) and instantaneous phase fluctuation *φ*(*t*) expressed in Eq. (1), the definition of the PSD [12], linearity and time-shifting properties of Fourier transform, we derived strictly the PSD of laser instantaneous phase fluctuation and frequency fluctuation, which can be expressed as:

From Eq. (5), the single-side-band phase noise can also be obtained with *L*(*f*) = *S _{φ}*(

*f*)/2 [13]. Equation (5) means that, at the low Fourier frequency domain, the PSD of the laser instantaneous phase fluctuation

*S*(

_{φ}*f*) would be larger than the PSD of the differential phase fluctuation

*S*

_{Δ}

*(*

_{φ}*f*), but at the high Fourier frequency domain, the former is smaller than the latter. However, the PSD of the laser instantaneous frequency fluctuation

*S*(

_{ν}*f*) is larger than the PSD of the differential frequency fluctuation

*S*

_{Δ}

*(*

_{ν}*f*) at any positive Fourier frequency. On the other hand, it is observed that if the differential phase and frequency fluctuation are normalized in 1 m delay fiber (

*τ*~5 ns),

*S*(

_{ν}*f*) would approximately equal to

*S*

_{Δ}

*(*

_{ν}*f*) at the Fourier frequency less than MHz level. In physics, the results can be explained that the frequency is the differential of the phase and the delay of the MI is equivalent to the differential operation for the phase. The conclusions are very important that the characterization of differential phase and instantaneous phase (laser phase) should be carefully distinguished but the instantaneous frequency can be replaced by the differential frequency sometimes in the practical engineering applications.

## 3. Experimental results and discussions

Firstly, to verify the correction of the phase demodulation of the setup, a demodulated test for a pre-set modulated phase by a LiNbO_{3} phase modulator is demonstrated. A narrow linewidth laser which is phase-modulated with the LiNbO_{3} phase modulator is used as LUT. Triangle waveform is selected to the modulation waveform for holding the frequency components as much as more. Hence the fact of the interference at the coupler is a subtraction between two triangle waveforms with delay time *τ* as shown in Fig. 2. The modulation period *T* needs to be twice of the delay time difference *τ*. Otherwise there are some constants in the interference fringe as shown in Fig. 2(a), accordingly the triangle wave cannot be demodulated. Once the triangle waveform is demodulated correctly as shown in Fig. 2(b), the demodulated amplitude would be twice the input modulated amplitude. Figure 3 shows the demodulated triangle phase amplitude and their waveforms at different modulated voltage amplitudes. The results confirmed the correction of the differential phase demodulation both in terms of waveform and amplitude. Figure 4 shows the time-domain interference fringes of the MI and the corresponding demodulated phase waveform at a fixed modulated voltage. The red curve and the black curve represent the results of two independent tests at different time, respectively. Despite the interference fringe changes at different time due to the environment variation, the demodulated phase would not change and hold the consistence with the input phase modulation. So the consistency and robustness of the proposed setup is verified.

Secondly, to demonstrate the noise measurement capacity of the setup, the phase and frequency noise of an external-cavity semiconductor laser (RIO ORION^{TM}) [14] with wavelength of 1551.7 nm and linewidth of about 2 kHz are measured. The PSD of the differential phase fluctuation normalized to 1 m delay fiber (*S*_{Δ}* _{φ}*(

*f*) @ 1m), the PSD of the differential phase fluctuation normalized to 1 m delay fiber (

*S*

_{Δ}

*(*

_{ν}*f*) @ 1m), the PSD of instantaneous frequency fluctuation

*S*(

_{ν}*f*), the PSD of the instantaneous phase fluctuation

*S*(

_{φ}*f*) and the laser phase noise

*L*(

*f*) are shown in Fig. 5. The data are very close to that given in the product datasheets or the typical data given in the website [14]. The curves clearly demonstrate the relations between these physical quantities as described above. At the focused frequency range (less than 1 MHz),

*S*

_{Δ}

*(*

_{ν}*f*) @ 1m approximately equals to

*S*(

_{ν}*f*),

*S*

_{Δ}

*(*

_{φ}*f*) @ 1m is much less than

*S*(

_{φ}*f*) and laser phase noise

*L*(

*f*) =

*S*(

_{φ}*f*)/2. So the usage of PSD of differential phase as the laser phase noise is not strictly correct and needs more careful definition and consideration.

From the PSD of the frequency fluctuation *S _{ν}*(

*f*), the linewidth at different observation time can be calculated. Figure 6 shows the linewidth calculated with the approximated model presented in [15] for different values of the integration bandwidth. The results indicate that the linewidth is very dependent on the integration bandwidth. Linewidth would increase with the increase of observation time (in other words, linewidth increase with the decrease of the lower limit of the integration bandwidth). It mainly results from the presence of the 1/

*f*type noise in the PSD of frequency fluctuation. At high frequency domain (>100 kHz), there is only white noise. And the minimum linewidth of about

^{α}*δν*

_{1}= 2 kHz is calculated. Meantime, the inset figure shows the linewidth of the same laser measured by the self-delay heterodyne (SDH) method [4] with heterodyne frequency of 80 MHz and optical fiber delay length of 45 km. The Lorentz fitted linewidth at −20 dB from the spectrum measured by the SDH method is about 51.7 kHz, so the laser linewidth is about

*δν*

_{2}= 2.6 kHz. And the fitted linewidth would not vary with the observation time. It indicates that the non-white noise components are not revealed in the Lorentz fitted results of SDH method. But the values are also conservative for white noise components and about 30% larger than the values calculated by the PSD of frequency fluctuation. This is because that the tail of the spectrum measured by SDH method is not taken into consideration in the Lorentz fitting process, resulting in the fitted value larger than the real value. Therefore, PSD of frequency fluctuation is recommended to completely describe the frequency noise behavior, and a specified linewidth value should be reported with the corresponding integration bandwidth or observation time.

## 4. Conclusion

A laser phase and frequency noise measurement technique by an unbalanced Michelson interferometer composed of a 3 × 3 optical fiber coupler and two Faraday rotator mirrors is proposed. In the method, the laser differential phase fluctuation accumulated by the interferometer delay time is demodulated directly at first. And then the phase and frequency noise is calculated by the PSD estimation for the differential phase. Also the concepts and differences of differential phase and frequency fluctuation PSDs, instantaneous phase and frequency fluctuation PSDs, and phase noise are defined strictly and discussed carefully. The method can obtain the noise features of a narrow linewidth laser without any specific assumptions or noise models. Meantime, the technique is used to characterize a narrow linewidth external-cavity semiconductor laser, which confirmed the correction of the method and revealed the fact that the linewidth would increase with the increase of observation time, and the Lorentz fitted linewidth measured by the SDH method only includes the contribution of the white noise components and would be larger than the real value.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (61405212).

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